2 Answers
2

Your guess would have been true if $dW$ were just an ordinary differential. But this is not the case. $dW$ behaves somewhat different from the ordinary differential, in that intuitively we have $dW \approx \sqrt{dt}$ unlike to ordinary case. That's why the Ito,s formula is so important.

Now let us make a heuristic calculation. Let $Z_t = f(S_t) = \log S_t$, where $f(x) = \log x$. Then by the Ito's formula, we have

The SDE can solved analytically and the solution is the Geometric Brownian Motion which has the form:
$$
S(t)=S(0)\exp\left(\left[\mu-\frac{\sigma^2}{2}\right]t+\sigma W(t)\right),
$$
where $(W(t))_{t\geq 0}$ is a Brownian motion.